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Extension

Extension
is an option for various polynomial and algebraic functions that specifies generators for the algebraic number field to be used.
  • For polynomial functions, Extension determines the algebraic number field in which the coefficients are assumed to lie.
  • The setting Extension->a specifies the field consisting of the rationals extended by the algebraic number a.
  • The must be exact numbers, and can involve radicals as well as Root and AlgebraicNumber objects.
  • Extension->Automatic specifies that any algebraic numbers that appear in the input should be included in the extension field.
  • For polynomial functions, the default setting Extension->None specifies that all coefficients are required to be rational. Any algebraic numbers appearing in input are treated like independent variables.
  • Extension includes both the and any algebraic numbers in the input.
Factor a polynomial over :
PolynomialGCD over the field generated by the algebraic numbers present in the coefficients:
Factor a polynomial over :
In[1]:=
Click for copyable input
Out[1]=
 
PolynomialGCD over the field generated by the algebraic numbers present in the coefficients:
In[1]:=
Click for copyable input
Out[1]=
By default, factorization is performed over the rationals:
This specifies the factorization should be done over the rationals extended by :
Here the factorization is done over the rationals extended by and I:
By default, PolynomialGCD treats algebraic numbers as independent variables:
This computes the GCD over the algebraic number field generated by the coefficients:
By default, Together treats algebraic numbers as independent variables:
With Extension->Automatic, Together recognizes algebraically dependent coefficients:
By default, the norm is computed in the field generated by the AlgebraicNumber object:
This computes the norm in the field in which the AlgebraicNumber object is represented:
This computes the norm in the field generated by :
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